Design of a microstrip Wilkinson power divider using a low pass filter with the particle swarm optimization algorithm

In this paper, a microstrip Wilkinson power divider (MWPD) based on particle swarm optimization (PSO) algorithm is designed, simulated, and fabricated using novel resonators. In addition, attenuators and open-ended stubs are incorporated to generate a broad cut-off band and reduce unwanted harmonics. The proposed power divider has a central frequency of 1 GHz. The performance of each used resonator is analyzed based on lumped-element circuit models.The L and C parameters of the equivalent circuit of the used resonators are predicted and optimized with the assistance of the PSO method. The subsequent phase was the fabrication of the proposed MWPD, after which its performance was evaluated in the light of the results obtained from the simulation. It was discovered that there was a high degree of concordance between the two. On the other hand, the fabricated circuit has several benefits, including a suitable S12 of − 3.15 dB, a high return loss of less than − 24 dB at the operating frequency, a compact size of 0.058 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\lambda}}_{{\varvec{g}}}$$\end{document}λg × 0.064 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{\lambda}}_{{\varvec{g}}}$$\end{document}λg, and the ability to remove undesired harmonics. The results show a high level of suppression of the unwanted harmonics (up to the 16th harmonic) and a great responsiveness in the passband, while having very low ripple. As a result, the proposed circuit may be used in a wide variety of electronic devices, such as radar transmitter and receiver circuits, and many other high-frequency systems.

ground structure is used to generate the meta-structure.Within the operating frequency range, it can suppress the second, third, and fourth harmonics in addition to realizing the impedance conversion function.
The article 13 presents the study and fabrication of a high-performance MWPD with a low insertion loss and a large suppression factor at 7 GHz.To form the planned MWPD, two open stubs are replaced with regular transmission lines.Furthermore, even and odd mode analysis was used to investigate the characteristics of the recommended stub.However, it has a convoluted design and a low FBW, which is a major drawback.Some of the presented research focuses on the geometrical structure of resonators.For example, rectangular resonators and U-shaped transmission lines are used in 14 .Another type of MWPD is presented in 15 where trapezoidal-shaped resonators are used.The dimensions of the presented structure are very small, and transmission line analysis is used for the mathematical analysis of the presented circuit.In 16 , triangular-shaped resonators were used to generate a large stop band bandwidth.This structure has suppressed the undesirable 16 harmonics.However, its insertion loss level is not adequate, and it also has minimal isolation between the output ports.
While coupled-line and resonator-based power dividers work well to reduce size and decrease harmonics, but they also increase the insertion loss parameter.Moreover, open-stub-based power dividers have a simple construction and reasonable performance; nevertheless, this approach is not very good at reducing size or suppressing harmonics 17 .
On the other hand, several researchers have focused their attention on the mathematical study of the power divider.As part of the design process, the neural network model and the LC-equivalent circuit model are utilized to predict the transmission zeros of the power divider provided in 18 .The required harmonics can be suppressed using these transmission zeros.The suggested power divider is now more effective, and it is all down to the neural network model's prediction of the major circuit parts.Also, in 19 , a surrogate neural network model was used to design a microstrip power divider and a low-pass filter (LPF).The suggested technique improves the performance and frequency response of a wide range of microwave devices.Their efficient design is a result of using the surrogate model of the proposed artificial neural network to determine the dimensions of the LPF and the power divider.However, the suggested structure has a restricted FBW and is difficult to produce due to its complexity.Ref. 20 presents yet another approach that may be utilized to study and estimate the parameters of a power divider.In this study, the PSO method was utilized to compute the values of the equivalent inductors and capacitors in the microstrip structure, which resulted in an extremely optimum result.In addition, it has proved to be very accurate.
In the rapidly evolving landscape of microwave and radio frequency (RF) engineering, the design and optimization of MWPDs play a pivotal role in improving the performance of communication systems.The demand for efficient and compact MWPDs has led researchers to explore innovative designs and optimization techniques.This scientific article delves into the complex field of power divider design, presenting a novel approach that combines new resonators with the PSO algorithm to achieve superior performance characteristics.The center frequency of the proposed small-size structure is 1 GHz, and it suppresses 16 unwanted harmonics.Also, its FBW is 148% and 173% under the conditions of − 15 dB return loss and − 3 dB insertion loss.The design, simulation, fabrication, and measurement of the proposed circuit involves several key steps.Here's a general outline of the process: (

Conventional structure of the MWPD
The design of the famous MWPD is shown in Fig. 1.It comprises two transmission-line sections with electrical lengths of 90º, characteristic impedances of 70.7 Ω, and one isolation resistor.The MWPD can be implemented using microstrip or stripline technology and is typically used in applications where high isolation and low insertion loss are required, such as in satellite communication systems, radar systems, and wireless networks.

Resonator design
In high-frequency circuits, resonators are used for various purposes such as filtering, tuning, and frequency stabilization.They can be used to select specific frequencies for transmission or reception, reject unwanted frequencies, or provide stable oscillations for signal generation.Resonators can be implemented using a number of different technologies, such as quartz crystals, ceramic materials, or LC (inductor-capacitor) circuits.Each technology has its advantages and limitations in terms of frequency range, stability, size, cost, and performance 1-3 .
Here, we have used new microstrip resonators to improve the performance of the conventional MWPD.In Figs.2a and b, the layout of the new resonator, and its LC equivalent circuit are shown, respectively.
Inductor and capacitor (LC) equivalent circuits may be designed for microstrip lines because of their shape.The performance of microstrip resonators can be easily analyzed with the help of this circuit, which produces a schematic diagram of the intended design 2 .
Therefore, the equivalent LC circuit of the proposed resonator is shown in Fig. 2b.Note that La is equal to Lb.To apply the LC model to microstrip lines, consider each one as a ground capacitor and a series inductor.It is possible to extrapolate the matching LC circuit from here.Accurate reproductions of transmission lines can be made using the following formulae 2,12, 21 : For w/h ≤ 1: where Z is the characteristic impedance and Vp is the phase velocity of the transmission line.The constant speed of light, c is equal to (120π) 377Ω, l is the length of the transmission line, w is the width of the line, and h is the thickness of the substrate.
Based on Fig. 2b, we may assume that the observed impedance is from V n is Z n and derive the transfer function as follows: In this design, L a = L b .The transfer function between Vo and Vi is calculated using Eqs. ( 7 and ( 8) as follows: The simulation results (EM analysis) of the proposed resonator in the structure of a basic Wilkinson power divider are shown in Fig. 3.At 6.6 GHz, the proposed resonator has a transmission zero.This resonator passes the input signal up to a frequency of about 3.2 GHz.Additionally, the S 11 parameter value in the pass band is less than − 10 dB.
The result in Fig. 3 shows that the resonators used create a transmission zero in the frequency response of the proposed power divider.However, we need to use resonators and other suppressors to improve the frequency response of the circuit to increase the bandwidth and eliminate the unwanted harmonics.Note that the proposed final circuit is placed on the main transmission line of the primary Wilkinson power divider.Indeed, we replace the primary transmission line of the circuit with a filter structure.

New suppressor
We modified the design of the proposed circuit, as shown in Fig. 4, to increase the stop band bandwidth and obtain a broad stop band.To provide a good stop band frequency response for the LPF, a new suppressor has been employed.Figure 4 depicts how several TZs are formed in the stop band as a result of the designed suppressor.
The LPF design is achieved by connecting the designed circuit to a suppressor.This suppressor is used to reduce the stop band S 12 parameter.The S 12 parameter should be reduced in the stop band of an LPF so that no signal passes through it.Our approach provides an innovative and straightforward solution to extend the stopping range.The LC equivalent circuit of the proposed suppressor is shown in Fig. 4b.
In Fig. 4b, Le to Lk and Cd to Cg are the inductances and capacitances of the transmission lines, respectively.In this design, the equivalent impedances at nodes Vq, and Vm are Zq, and Zm, respectively.So, using KCL and KVL relations between circuit elements, it can be written as follows: (1)  www.nature.com/scientificreports/In this circuit L e = L k .The transfer function between Vo and Vi can be calculated using Eqs. ( 10 to (18) as follows: The simulation result (EM analysis) of the proposed suppressor in the structure of a basic Wilkinson power divider is shown in Fig. 5.As expected, it produced two transmission zeros at around 11.4 and 20.8 GHz. (10) Result of the simulation of the S 12 and S 11 parameters of the proposed suppressor.
The layout module of the Advanced Design System (ADS) software models the resonator and suppressor shown in Figs.2a and 4a using the Rogers 5880 substrate (h = 20 mil, tan d = 0.0009, and ε r = 2.2).Consequently, for each section of the microstrip line, we know the dimensions of the resonator and suppressor, namely their length (l) and width (w).In contrast, the value of the inductor and capacitor corresponding to this line may be easily determined using Eqs.(1) to (4).This is done for each transmission line based on the equivalent circuit shown in Figs.2b and 4b.This is because all the important criteria are available.Equations (1) to (4) are used to determine the initial values of these capacitors and inductors in Table 1.
Figures 3 and 5 demonstrate that the return losses of the proposed circuit are unsuitable in both the passband and the cut-off band, and that their cut-off bandwidth is limited.Therefore, mathematical optimization algorithms such as PSO can be used to calculate the most optimal result.

Estimation of parameters and optimization
PSO has been used to determine optimal design parameters because it performs well in designing microwave circuits 22 .Equation (20) shows the best design objective function.
To design an ultra-wide bandwidth power divider, we need to use a two-part Eq. (20).At the operating frequencies of the power divider, the insertion loss should reach its maximum value (-3 dB) in the objective function while minimizing the input/output return loss.As a result, the optimal parameters of the power divider are those found by applying these requirements.Optimization techniques are the best choice to determine them.
PSO is an advanced algorithm in the field of cumulative intelligence, introduced in 23 .In recent years, the PSO algorithm has received much attention due to its simplicity and ease of use [24][25][26] .
The PSO algorithm is a computational algorithm developed based on the group behavior of insects and birds in search of food.This algorithm is used to solve complex and non-linear optimization problems.Particles in the PSO algorithm move around their current location in the search space at their own speed.They then update their speed and location based on their performance and the performance of the best particle so far, as well as the best location they have found so far.This process continues iteratively until they reach the optimal point in the search space.
By using the PSO algorithm, complex and non-linear optimization problems can be solved.The search process requires two memories for each particle, one of which is used to maintain the optimal position of the particle.The particles decide on their next steps based on the data they extract from these memories.In each iteration, the velocity and position of each particle are updated based on the best absolute and local solutions available 27 .By adding the velocity of the particle to its current position, the location of the population may be determined.
X j (i) indicates the position of particle j, V j (i) indicates the velocity and i indicates the number of times the velocity variable is repeated.The velocity can be determined using the following equation: V j (i) is the i-th component of the velocity of particle j, r 1 , and r 2 are uniform random values distributed in the interval (0,1), and based on experiments 27 , the parameters c 1 and c 2 represent individual and group learning components.
The particle's local best position, P best,j , and global best position, G best are also given.θ(i) is the inertial weight used to adjust the particle velocity in controlled laboratory experiments.From 28 , we can derive θ(i): θ min is the minimum number of iterations in the algorithm and θ max is the maximum number of iterations.θ min and θ max are the minimum and maximum inertia weights, respectively.Experimental results often show that θ min = 0.4 and θ max = 0.7 are optimal 29 .
The design parameters for this work were optimized using the PSO technique with the values given in Table 2.
Once the overall design of the proposed resonator and suppressor has been finalized, the next step is to improve the circuit's performance by finding the optimal values for the inductors and capacitors.The optimization ( 20) The transmission zero (T Z1 ) for the proposed circuit is shown in Fig. 3 to be approximately 6.6 GHz, whereas the T Z1 for the arrangement is approximately 6.62 GHz.The findings of the layout and the presented circuit are in good agreement.Also, the calculated T Z2 and T Z3 were almost equal to the simulation results.In other words, placing the proposed resonator and suppressor together will produce a semi-LPF function.
Figures 7a and b respectively show the simulation results of the S-parameters of the resonator and suppressor circuits with optimized values.These results have been compared with the simulation results obtained using the initial values.As it is known, when the circuit elements have optimized values, a more suitable passband and cut-off band have been created.According to Fig. 7a, in the optimized mode, the S 12 response is sharper and has (26)  T z3 = 20.75GHzwww.nature.com/scientificreports/created a wide cut-off band up to − 25 GHz with a level below − 24 dB.Also, the S 11 level is lower in the pass band (− 22 dB).The results in Fig. 7b also show that the suppressor circuit with the best values from frequency 11.2 to 23.3 GHz has an S 12 value of less than − 20 dB, which is much better than the circuit with the initial values.The S 11 parameter in the cut-off band has a value close to zero.Thanks to the optimisation process, the developed circuit really has a superior response.

Final structure of the designed MWPD
In order to achieve ultra-wide bandwidth and a suitable operating frequency, new resonators have been used symmetrically and inverted on both sides of the main transmission line of the circuit.A suppressor is also used to weaken unwanted signals.In the present design, the coupling effects of nearby lines are ineffective and are ignored in the calculations.On the other hand, the small stubs at the end have been used to create a wide cutting band along the transmission line.
The final design of the MWPD is shown in Fig. 8.The important point and one of the innovations used in this layout is the use of two 47-Ω SMD resistors instead of the conventional 100-Ω isolation resistor in the MWPD structure.Also, the dimensions of the resonators, suppressors, and open-ended stubs in millimeters are shown in Fig. 8.
Due to the positioning of the stubs between the output ports, it is not feasible to insert a resistor in the redesigned arrangement depicted in Fig. 8. Consequently, it is necessary to employ two resistors in order to separate the output signals from one another.
Figure 9 displays the simulation results of S 23 for various nominal levels of isolation resistance.From the simulation results, it can be observed that the highest level of isolation between the two output ports is achieved when the resistance is set to 47-Ω at the center frequency.

Fabrication of the proposed MWPD
After designing and simulating the proposed MWPD and calculating its parameters, the designed power divider is fabricated.This designed circuit is fabricated on a Rogers 5880 substrate with a thickness of 20 mil, a loss of 0.0009, and ε r = 2.2. Figure 10 shows the final circuit.In this structure, two 47-Ω resistors are used, following the  www.nature.com/scientificreports/conventional structure of the MWPD, which improves the thermal conductivity of the circuit.It should be noted that ADS software was used for the design and simulation, and the fabricated power divider was measured by the B8510 network analyzer.Also, the fabricated circuit is shown in Fig. 10.

Results and discussion
This novel power divider features a small size (12.9 mm × 14.2 mm, or 0.058 g × 0.064 g ), excellent bandwidth, ideal return loss, and an operating frequency of 1 GHz.Also, the results of the simulation and measurement of parameters S 11 and S 12 are shown in Fig. 11a. Figure 11a shows that in the range of 0.32 GHz to 1.8 GHz, the proposed structure has S 11 with an attenuation level below − 15 dB.Therefore, the FBW in the range where the return loss is less than − 15 dB is equal to 148% and in the range where the insertion loss is − 3 dB is equal to 173%, which is very ideal.Also, the unwanted harmonics of this circuit have been removed, up to 16 unwanted harmonics with a level below − 19.The presented results show that the operating band is extremely wide.At the operating frequency (1 GHz), the value of S 12 is approximately − 3.15 dB, which is very suitable, and the passband of the power divider is relatively flat and does not fluctuate much.
Figure 11b shows the results of the simulation and measurement of the output return loss (S 22 ) and isolation between the parameters of two output ports (S 23 ).The obtained measurements show that the value of S 22 is approximately − 21 dB and that of S 23 is approximately − 19 dB at the operating frequency of the designed MWPD.As a result, the output port of the power divider exhibits zero return loss, and the operational performance of the power divider ensures complete isolation between the two output ports.
Figures 11a and b show that the simulation and measurement results are very close to each other with only small differences.Possible reasons for these differences include connection losses, substrate characteristics, and the soldering of isolation resistors.Therefore, the fabricated circuit is quite practical and effective.
The current density distribution is depicted in Figs.12a and b, respectively, for frequencies of 1 and 6 GHz.At a frequency of 1 GHz, as is generally accepted, we have an appropriate distribution of the electromagnetic field in the output ports, and the signal is allowed to pass.However, at a frequency of 6 GHz, which is in the cut-off band of the filter, the designed resonators and stubs prevent the signal from passing, and nothing at all is allowed to reach the output.
The final step was to compare the MWPD obtained using the filter circuit and the PSO optimization algorithm with other similar works that have been published.The results of this comparison are presented in Table 4.
As can be seen, the circuit presented here is better in terms of FBW than all those mentioned in Table 4 and has the highest value.The S 12 parameter has the most ideal conditions compared to the rest of the articles, and the dimensions of the designed circuit are relatively small and suitable considering that the circuit arrangement is in the Wilkinson form.In addition, this circuit has removed up to 16 disturbing harmonics.The designed MWPD shows great promise for various applications within the realm of microwave engineering, including radar systems, phased array antennas, aerospace, and defense.
The main novelty of this work is the use of two isolation resistors and a new suppressor.Furthermore, for the first time, the PSO algorithm has been used to calculate the LC equivalent circuit parameters of an MWPD to achieve improved performance in terms of insertion loss, input return loss, and isolation compared to traditional designs.

Conclusion
In this paper, a compact 1 GHz MWPD with new resonators and suppressors was proposed.The incorporation of these resonators has introduced a fresh perspective on geometric configurations, resulting in tangible improvements in key performance metrics.To obtain the parameters of the presented circuit, the PSO method was used.Also, in this article, two 47-Ω isolation resistors are used between the output ports.This innovation increased FBW by 148% and 173%, improved S 12 by − 3.15 dB, and reduced S 11 by − 24 dB to address critical challenges in traditional power divider designs.The dimensions of this circuit are only 0.058 g × 0.064 g .In addition, the presented structure suppresses the unwanted 16th harmonic, so it can be used in many RF and telecommunication systems.
1) Problem formulation and specification: MWPD specifications and requirements definition.Determining the parameters of open stubs and new resonators in terms of their effect on the overall performance of the power divider (2) Initial circuit design: Designing the initial circuit based on the MWPD topology that incorporates new resonators and open stubs.Definition of the circuit elements, dimensions, and related materials for the resonators (3) PSO optimization: Implementation of the PSO optimization algorithm to fine-tune the LPF parameters.(4) Simulation: An electromagnetic simulation tool (ADS software) is used to simulate the performance of the designed MWPD.The validity of the optimized design is confirmed against the specified requirements and objectives.(5) Refinement and iteration: Based on the simulation results, the design is modified iteratively, adjusting the parameters and optimizing until the desired performance is reached.(6) PCB Layout and Fabrication: Translation of the optimized circuit design to the printed circuit board (PCB) design.(7) Measurement Setup: Setup a measurement environment with appropriate RF test equipment, including network analyzers and power meters.(8) Data Analysis and Comparison: Analyzing the measured data and comparing it with simulated results to validate the accuracy of the design.

Figure 3 .
Figure 3. Simulation results of the of the S 12 and S 11 parameters of the proposed resonator.

Figure 4 .
Figure 4. (a) Proposed suppressor (b) LC equivalent circuit of the proposed suppressor.

Figure 7 .
Figure 7. Simulation results of the scattering parameters with initial and optimized values for (a) designed resonator, and (b) designed suppressor.

Figure 8 .
Figure 8. Final design of the proposed MWPD.

Figure 9 .
Figure 9. Simulation results of S 23 for different values of isolation resistance.

Table 1 .
Initial values of equivalent inductors and capacitors (unit: L: nH; C: pF).

Table 3 .
Values for L and C obtained from the power divider provided based on the PSO method (unit: L: nH; C: pF).

Table 4 .
Comparison of the designed power divider with other works.